A Lagrangian approach to droplet condensation in turbulent clouds Rutger IJzermans, Michael W. Reeks School of Mechanical & Systems Engineering Newcastle University, United Kingdom Ryan Sidin Department of Mechanical Engineering University of Twente, the Netherlands

Objective and motivation Research question:How does turbulence influence condensational growth of droplets? Application:Rain initiation in atmospheric clouds Objectives:- Gain understanding of rain initiation process, from cloud condensation nuclei to rain droplets - Elucidate role of turbulent macro-scales and micro-scales on condensation of droplets in clouds

Background: scales in turbulent clouds Turbulence: Large scales:L 0 ~ 100 m,  0 ~ 10 3 s,u 0 ~ 1 m/s, Small scales:  ~ 1 mm,  k ~ 0.04 s,u k ~ m/s. Droplets:Radius:Inertia:Settling velocity: Formation:r d ~ m,St =  d /  k ~ 2 × 10 -6,v T /u k ~ 3 × Microscales:r d ~ m,St =  d /  k ~ 0.02,v T /u k ~ 0.3 Rain drops:r d ~ m,St =  d /  k ~ 200,v T /u k ~ 3000 COLLISIONS / COALESCENCE CONDENSATION Collisions / coalescence process vastly enhanced if droplet size distribution at micro-scales is broad

Classic theory (Twomey (1959); Shaw (2003)): Fluid parcel, filled with many droplets of different sizes Droplet size distribution at microscales If parcel rises, temperature decreases due to adiabatic expansion, and supersaturation s increases: P roblem: Droplet size distribution in reality (experiments) becomes broader O(  ) Droplet growth is given by: or: Size distribution PDF(r d ) becomes narrower in time! Twomey’s fluid parcel approximation is not allowed in turbulence

Cloud turbulence modelled by kinematic simulation: All relevant flow scales can be incorporated by choosing k n of appropriate length Turbulent energy spectrum required as input Numerical model for condensation in cloud Ideally, Direct Numerical Simulation of: Velocity and pressure fields (Navier-Stokes) Supersaturation and temperature fields Computationally too expensive: O(L 0 /  ) 3 ~ cells - State-of-the-art DNS: modes, L 0 ~ 70cm (Lanotte et al., J. Atm. Sci. (2008))

Energy spectrum in wavenumber space

Full condensation model - rate-of-change of droplet mass m d : - rate-of-change of mixture temperature T: - rate-of-change of supersaturation s: Droplet modelled as passive tracer, contained within a moving air parcel: Along its trajectory (Lagrangian): Mixture of air & water vapour Latent heat release Adiabatic cooling Vapour depletion volume V p

Simplified condensation model Track droplets as passive tracers: Rate-of-change of droplet mass m d : Temperature T and supersaturation s are assumed to depend on adiabatic cooling only: Air & water vapor

Typical supersaturation profile Imposed mean temperature and supersaturation profiles: Focus on regions where supersaturation is close to zero

Determine droplet size distribution: Droplet population (N d =8000) initially randomly distributed in a plane at time t = t e and height z e : Droplet trajectories traced backward in time: t = t e  0 At t = 0 a monodisperse distribution is assumed: r d (0) = r 0 = m Condensation model equations are integrated forward in time to obtain droplet size distribution in the plane at t = t e Computational strategy

Results: dispersion in 3D KS-flow field 1-particle statistics: Short times: ~ t 2 Long times: ~ t 2-particle statistics: Slope = 4.5, similar to [Thomson & Devenish, J.F.M. 2005] Flight of 2 particles initially separated by distance r 0 =  : In agreement with Taylor (1921)

Time evolution of droplet position and size z e = 1355 m ; t e =100 s ; size of sampling area = 1 x 1 cm 2 Backward tracing: t = t e  0 Forward tracing: t = 0  t e

Droplet evaporation in regions where s < 0 Number of droplets at various altitudes: Rapid initial evaporation Forward tracing: t = 0  t c

Droplet radius distributions in time Temporal evolution of radius distribution function (z 0 = 1355 m):

Radius distributions after t e = 100 s Influence of measurement altitude: (size of sampling area L = 500 m) Influence of sampling area width: (z e = 1350 m)

Effect of different scales in turbulence Droplet radius distribution in flow with: - Only large scales included (n=1-10) - Only small scales included (n= ) - Wide range of scales included (n=1-200) z e = 1350m, t e = 100s, L = 0.01m

Results for two-way coupled model Eulerian evolution of droplet size distribution for n d = (5 η) -3 = 8.0 x 10 6 m -3 : z e = 1350 m z e = 1380m

Results two-way coupled model: interpretation Saturation of droplet radius distribution function follows from a balance between: - Adiabatic expansion (“forcing”) - Vapour depletion (“damping” with time scale  s ) - Latent heat release (“damping” with time scale  L ) Equation for supersaturation s is: with: This can be rewritten into:

Results two-way coupled model: interpretation Relative importance of the two damping terms,  s /  L, as a funciton of temperature: Dependence of vapour depletion time scale  s on droplet radius r d and on droplet number density n d :

Results two-way coupled model t e = 100s, z e = 1350m, L=500m Influence of droplet number density n d :

Results two-way coupled model t e =100s, z e = 1350m, n d =(2  ) -3 = x 10 9 m -3 Influence of length of the sampling area L:

Conclusions Droplet size distribution may become broader during condensation: - Large scales of turbulent motion responsible for transport of droplets to different regions of the flow, with different supersaturations - Small scales of turbulent motion responsible for local mixing of large and small droplets Broad droplet size distribution observed both in simplified condensation model and in two-way coupled condensation model Broadening of droplet size distribution enhanced by: - Higher flow velocities (more vigourous turbulence) - Lower droplet number density - Lower surrounding temperatures